Equations of the form (REF ) (and also variants with convection) appear in numerous applications. We selectively mention reservoir simulation, sedimentation processes, and traffic flow in the local case [1]}, [2]}, [3]}; cardiac electrophysiology and semiconductor growth in the nonlocal case [4]}, [5]}; and flows of fluids through porous media, mathematical finance, information theory, and phase transitions in both cases [6]}, [7]}, [8]}, [9]}, [10]}. Important examples are strongly degenerate Stefan problems (cf. [11]}, [12]}) with \(\varphi (u)=\max \lbrace 0,u-L\rbrace \) , \(L>0\) , and the full range of porous medium and fast diffusion equations (cf. [7]}) with \(\varphi (u)=u|u|^{m-1}\) , \(m>0\) . The class of operators \(\mathfrak {L}^{\sigma ,\mu }\)  coincides with the generators of symmetric Lévy processes [14]}, [15]}, [16]}. This includes e.g. the Laplacian \(\Delta \) , fractional Laplacians \(-(-\Delta )^{\frac{\alpha }{2}}\) , \(\alpha \in (0,2)\) , relativistic Schrödinger type operators \(m^\alpha I-(m^2I-\Delta )^{\frac{\alpha }{2}}\) , \(\alpha \in (0,2)\)  and \(m>0\) , tempered stable processes [8]}, and even discretizations of all of these. Since \(\sigma \)  and \(\mu \)  may be degenerate or even identically zero, equation (REF ) can be purely nonlocal, purely local, or a combination.
